Dr. William Murray’s Differential Equations course will help you solve problems from all over biology, physics, chemistry, and engineering. Dr. Murray demonstrates his extensive teaching experience with easy to understand theory and a wide array of examples and helpful tips. Each lesson also includes several completely worked out practice problems like the ones you will see on homework and tests. Sample topics include linear and separable equations, second order equations, series solutions, systems of equations, and Laplace transforms. Dr. Murray received his Ph.D from UC Berkeley, his BS from Georgetown University, and has been teaching in the university setting for 10+ years.

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I. First-Order Equations
  Linear Equations 1:07:21
   Intro 0:00 
   Lesson Objectives 0:19 
   How to Solve Linear Equations 2:54 
    Calculate the Integrating Factor 2:58 
    Changes the Left Side so We Can Integrate Both Sides 3:27 
    Solving Linear Equations 5:32 
   Further Notes 6:10 
    If P(x) is Negative 6:26 
    Leave Off the Constant 9:38 
    The C Is Important When Integrating Both Sides of the Equation 9:55 
   Example 1 10:29 
   Example 2 22:56 
   Example 3 36:12 
   Example 4 39:24 
   Example 5 44:10 
   Example 6 56:42 
  Separable Equations 35:11
   Intro 0:00 
   Lesson Objectives 0:19 
    Some Equations Are Both Linear and Separable So You Can Use Either Technique to Solve Them 1:33 
    Important to Add C When You Do the Integration 2:27 
   Example 1 4:28 
   Example 2 10:45 
   Example 3 14:43 
   Example 4 19:21 
   Example 5 27:23 
  Slope & Direction Fields 1:11:36
   Intro 0:00 
   Lesson Objectives 0:20 
    If You Can Manipulate a Differential Equation Into a Certain Form, You Can Draw a Slope Field Also Known as a Direction Field 0:23 
    How You Do This 0:45 
   Solution Trajectories 2:49 
    Never Cross Each Other 3:44 
    General Solution to the Differential Equation 4:03 
    Use an Initial Condition to Find Which Solution Trajectory You Want 4:59 
   Example 1 6:52 
   Example 2 14:20 
   Example 3 26:36 
   Example 4 34:21 
   Example 5 46:09 
   Example 6 59:51 
  Applications, Modeling, & Word Problems of First-Order Equations 1:05:19
   Intro 0:00 
   Lesson Overview 0:38 
    Mixing 1:00 
    Population 2:49 
    Finance 3:22 
    Set Variables 4:39 
    Write Differential Equation 6:29 
    Solve It 10:54 
    Answer Questions 11:47 
   Example 1 13:29 
   Example 2 24:53 
   Example 3 32:13 
   Example 4 42:46 
   Example 5 55:05 
  Autonomous Equations & Phase Plane Analysis 1:01:20
   Intro 0:00 
   Lesson Overview 0:18 
    Autonomous Differential Equations Have the Form y' = f(x) 0:21 
    Phase Plane Analysis 0:48 
    y' < 0 2:56 
    y' > 0 3:04 
    If we Perturb the Equilibrium Solutions 5:51 
    Equilibrium Solutions 7:44 
    Solutions Will Return to Stable Equilibria 8:06 
    Solutions Will Tend Away From Unstable Equilibria 9:32 
    Semistable Equilibria 10:59 
   Example 1 11:43 
   Example 2 15:50 
   Example 3 28:27 
   Example 4 31:35 
   Example 5 43:03 
   Example 6 49:01 
II. Second-Order Equations
  Distinct Roots of Second Order Equations 28:44
   Intro 0:00 
   Lesson Overview 0:36 
    Linear Means 0:50 
    Second-Order 1:15 
    Homogeneous 1:30 
    Constant Coefficient 1:55 
    Solve the Characteristic Equation 2:33 
    Roots r1 and r2 3:43 
    To Find c1 and c2, Use Initial Conditions 4:50 
   Example 1 5:46 
   Example 2 8:20 
   Example 3 16:20 
   Example 4 18:26 
   Example 5 23:52 
  Complex Roots of Second Order Equations 31:49
   Intro 0:00 
   Lesson Overview 0:15 
    Sometimes The Characteristic Equation Has Complex Roots 1:12 
   Example 1 3:21 
   Example 2 7:42 
   Example 3 15:25 
   Example 4 18:59 
   Example 5 27:52 
  Repeated Roots & Reduction of Order 43:02
   Intro 0:00 
   Lesson Overview 0:23 
    If the Characteristic Equation Has a Double Root 1:46 
    Reduction of Order 3:10 
   Example 1 7:23 
   Example 2 9:20 
   Example 3 14:12 
   Example 4 31:49 
   Example 5 33:21 
  Undetermined Coefficients of Inhomogeneous Equations 50:01
   Intro 0:00 
   Lesson Overview 0:11 
    Inhomogeneous Equation Means the Right Hand Side is Not 0 Anymore 0:21 
    First Solve the Inhomogeneous Equation 1:04 
    Find a Particular Solution to the Inhomogeneous Equation Using Undetermined Coefficients 2:03 
    g(t) vs. Guess for ypar 2:42 
    If Any Term of Your Guess for ypar Looks Like Any Term of yhom 5:07 
   Example 1 7:54 
   Example 2 15:25 
   Example 3 23:45 
   Example 4 33:35 
   Example 5 42:57 
  Inhomogeneous Equations: Variation of Parameters 49:22
   Intro 0:00 
   Lesson Overview 0:31 
    Inhomogeneous vs. Homogeneous 0:47 
    First Solve the Inhomogeneous Equation 1:17 
    Notice There is No Coefficient in Front of y'' 1:27 
    Find a Particular Solution to the Inhomogeneous Equation Using Variation of Parameters 2:32 
    How to Solve 4:33 
    Hint on Solving the System 5:23 
   Example 1 7:27 
   Example 2 17:46 
   Example 3 23:14 
   Example 4 31:49 
   Example 5 36:00 
III. Series Solutions
  Review of Power Series 57:38
   Intro 0:00 
   Lesson Overview 0:36 
    Taylor Series Expansion 0:37 
    Maclaurin Series 2:36 
    Common Maclaurin Series to Remember From Calculus 3:35 
    Radius of Convergence 7:58 
    Ratio Test 12:05 
   Example 1 15:18 
   Example 2 20:02 
   Example 3 27:32 
   Example 4 39:33 
   Example 5 45:42 
  Series Solutions Near an Ordinary Point 1:20:28
   Intro 0:00 
   Lesson Overview 0:49 
    Guess a Power Series Solution and Calculate Its Derivatives, Example 1 1:03 
    Guess a Power Series Solution and Calculate Its Derivatives, Example 2 3:14 
    Combine the Series 5:00 
    Match Exponents on x By Shifting Indices 5:11 
    Match Starting Indices By Pulling Out Initial Terms 5:51 
    Find a Recurrence Relation on the Coefficients 7:09 
   Example 1 7:46 
   Example 2 19:10 
   Example 3 29:57 
   Example 4 41:46 
   Example 5 57:23 
   Example 6 69:12 
  Euler Equations 24:42
   Intro 0:00 
   Lesson Overview 0:11 
    Euler Equation 0:15 
    Real, Distinct Roots 2:22 
    Real, Repeated Roots 2:37 
    Complex Roots 2:49 
   Example 1 3:51 
   Example 2 6:20 
   Example 3 8:27 
   Example 4 13:04 
   Example 5 15:31 
   Example 6 18:31 
  Series Solutions 1:20:26
   Intro 0:00 
   Lesson Overview 0:13 
    Singular Point 1:17 
   Definition: Pole of Order n 1:58 
    Pole Of Order n 2:04 
    Regular Singular Point 3:25 
   Solving Around Regular Singular Points 7:08 
    Indical Equation 7:30 
    If the Difference Between the Roots is An Integer 8:06 
    If the Difference Between the Roots is Not An Integer 8:29 
   Example 1 8:47 
   Example 2 14:57 
   Example 3 25:40 
   Example 4 47:23 
   Example 5 69:01 
IV. Laplace Transform
  Laplace Transforms 41:52
   Intro 0:00 
   Lesson Overview 0:09 
    Laplace Transform of a Function f(t) 0:18 
    Laplace Transform is Linear 1:04 
   Example 1 1:43 
   Example 2 18:30 
   Example 3 22:06 
   Example 4 28:27 
   Example 5 33:54 
  Inverse Laplace Transforms 47:05
   Intro 0:00 
   Lesson Overview 0:09 
    Laplace Transform L{f} 0:13 
    Run Partial Fractions 0:24 
   Common Laplace Transforms 1:20 
   Example 1 3:24 
   Example 2 9:55 
   Example 3 14:49 
   Example 4 22:03 
   Example 5 33:51 
  Laplace Transform Initial Value Problems 45:15
   Intro 0:00 
   Lesson Overview 0:12 
    Start With Initial Value Problem 0:14 
    Take the Laplace Transform of Both Sides of the Differential Equation 0:37 
    Plug in the Identities 1:20 
    Take the Inverse Laplace Transform to Find y 2:40 
    Example 1 4:15 
    Example 2 11:30 
    Example 3 17:59 
    Example 4 24:51 
    Example 5 36:05 
V. Review of Linear Algebra
  Review of Linear Algebra 57:30
   Intro 0:00 
   Lesson Overview 0:41 
    Matrix 0:54 
    Determinants 4:45 
   3x3 Determinants 5:08 
   Eigenvalues and Eigenvectors 7:01 
    Eigenvector 7:48 
    Eigenvalue 7:54 
   Lesson Overview 8:17 
    Characteristic Polynomial 8:47 
    Find Corresponding Eigenvector 9:03 
   Example 1 10:19 
   Example 2 16:49 
   Example 3 20:52 
   Example 4 25:34 
   Example 5 35:05 
VI. Systems of Equations
  Distinct Real Eigenvalues 59:26
   Intro 0:00 
   Lesson Overview 1:11 
   How to Solve Systems 2:48 
    Find the Eigenvalues and Their Corresponding Eigenvectors 2:50 
    General Solution 4:30 
    Use Initial Conditions to Find c1 and c2 4:57 
   Graphing the Solutions 5:20 
    Solution Trajectories Tend Towards 0 or ∞ Depending on Whether r1 or r2 are Positive or Negative 6:35 
    Solution Trajectories Tend Towards the Axis Spanned by the Eigenvector Corresponding to the Larger Eigenvalue 7:27 
   Example 1 9:05 
   Example 2 21:06 
   Example 3 26:38 
   Example 4 36:40 
   Example 5 43:26 
   Example 6 51:33 
  Complex Eigenvalues 1:03:54
   Intro 0:00 
   Lesson Overview 0:47 
    Recall That to Solve the System of Linear Differential Equations, We find the Eigenvalues and Eigenvectors 0:52 
    If the Eigenvalues are Complex, Then They Will Occur in Conjugate Pairs 1:13 
   Expanding Complex Solutions 2:55 
    Euler's Formula 2:56 
    Multiply This Into the Eigenvector, and Separate Into Real and Imaginary Parts 1:18 
   Graphing Solutions From Complex Eigenvalues 5:34 
   Example 1 9:03 
   Example 2 20:48 
   Example 3 28:34 
   Example 4 41:28 
   Example 5 51:21 
  Repeated Eigenvalues 45:17
   Intro 0:00 
   Lesson Overview 0:44 
    If the Characteristic Equation Has a Repeated Root, Then We First Find the Corresponding Eigenvector 1:14 
    Find the Generalized Eigenvector 1:25 
   Solutions from Repeated Eigenvalues 2:22 
    Form the Two Principal Solutions and the Two General Solution 2:23 
    Use Initial Conditions to Solve for c1 and c2 3:41 
   Graphing the Solutions 3:53 
   Example 1 8:10 
   Example 2 16:24 
   Example 3 23:25 
   Example 4 31:04 
   Example 5 38:17 
VII. Inhomogeneous Systems
  Undetermined Coefficients for Inhomogeneous Systems 43:37
   Intro 0:00 
   Lesson Overview 0:35 
    First Solve the Corresponding Homogeneous System x'=Ax 0:37 
   Solving the Inhomogeneous System 2:32 
    Look for a Single Particular Solution xpar to the Inhomogeneous System 2:36 
    Plug the Guess Into the System and Solve for the Coefficients 3:27 
    Add the Homogeneous Solution and the Particular Solution to Get the General Solution 3:52 
   Example 1 4:49 
   Example 2 9:30 
   Example 3 15:54 
   Example 4 20:39 
   Example 5 29:43 
   Example 6 37:41 
  Variation of Parameters for Inhomogeneous Systems 1:08:12
   Intro 0:00 
   Lesson Overview 0:37 
    Find Two Solutions to the Homogeneous System 2:04 
    Look for a Single Particular Solution xpar to the inhomogeneous system as follows 2:59 
   Solutions by Variation of Parameters 3:35 
   General Solution and Matrix Inversion 6:35 
    General Solution 6:41 
    Hint for Finding Ψ-1 6:58 
   Example 1 8:13 
   Example 2 16:23 
   Example 3 32:23 
   Example 4 37:34 
   Example 5 49:00 
VIII. Numerical Techniques
  Euler's Method 45:30
   Intro 0:00 
   Lesson Overview 0:32 
    Euler's Method is a Way to Find Numerical Approximations for Initial Value Problems That We Cannot Solve Analytically 0:34 
    Based on Drawing Lines Along Slopes in a Direction Field 1:18 
   Formulas for Euler's Method 1:57 
   Example 1 4:47 
   Example 2 14:45 
   Example 3 24:03 
   Example 4 33:01 
   Example 5 37:55 
  Runge-Kutta & The Improved Euler Method 41:04
   Intro 0:00 
   Lesson Overview 0:43 
    Runge-Kutta is Know as the Improved Euler Method 0:46 
    More Sophisticated Than Euler's Method 1:09 
    It is the Fundamental Algorithm Used in Most Professional Software to Solve Differential Equations 1:16 
    Order 2 Runge-Kutta Algorithm 1:45 
   Runge-Kutta Order 2 Algorithm 2:09 
   Example 1 4:57 
   Example 2 10:57 
   Example 3 19:45 
   Example 4 24:35 
   Example 5 31:39 
IX. Partial Differential Equations
  Review of Partial Derivatives 38:22
   Intro 0:00 
   Lesson Overview 1:04 
    Partial Derivative of u with respect to x 1:37 
    Geometrically, ux Represents the Slope As You Walk in the x-direction on the Surface 2:47 
   Computing Partial Derivatives 3:46 
    Algebraically, to Find ux You Treat The Other Variable t as a Constant and Take the Derivative with Respect to x 3:49 
    Second Partial Derivatives 4:16 
    Clairaut's Theorem Says that the Two 'Mixed Partials' Are Always Equal 5:21 
   Example 1 5:34 
   Example 2 7:40 
   Example 3 11:17 
   Example 4 14:23 
   Example 5 31:55 
  The Heat Equation 44:40
   Intro 0:00 
   Lesson Overview 0:28 
    Partial Differential Equation 0:33 
    Most Common Ones 1:17 
    Boundary Value Problem 1:41 
   Common Partial Differential Equations 3:41 
    Heat Equation 4:04 
    Wave Equation 5:44 
    Laplace's Equation 7:50 
   Example 1 8:35 
   Example 2 14:21 
   Example 3 21:04 
   Example 4 25:54 
   Example 5 35:12 
  Separation of Variables 57:44
   Intro 0:00 
   Lesson Overview 0:26 
    Separation of Variables is a Technique for Solving Some Partial Differential Equations 0:29 
   Separation of Variables 2:35 
    Try to Separate the Variables 2:38 
    If You Can, Then Both Sides Must Be Constant 2:52 
    Reorganize These Intro Two Ordinary Differential Equations 3:05 
   Example 1 4:41 
   Example 2 11:06 
   Example 3 18:30 
   Example 4 25:49 
   Example 5 32:53 
  Fourier Series 1:24:33
   Intro 0:00 
   Lesson Overview 0:38 
    Fourier Series 0:42 
    Find the Fourier Coefficients by the Formulas 2:05 
   Notes on Fourier Series 3:34 
    Formula Simplifies 3:35 
    Function Must be Periodic 4:23 
   Even and Odd Functions 5:37 
    Definition 5:45 
    Examples 6:03 
   Even and Odd Functions and Fourier Series 9:47 
    If f is Even 9:52 
    If f is Odd 11:29 
   Extending Functions 12:46 
    If We Want a Cosine Series 14:13 
    If We Wants a Sine Series 15:20 
   Example 1 17:39 
   Example 2 43:23 
   Example 3 51:14 
   Example 4 61:52 
   Example 5 71:53 
  Solution of the Heat Equation 47:41
   Intro 0:00 
   Lesson Overview 0:22 
   Solving the Heat Equation 1:03 
   Procedure for the Heat Equation 3:29 
    Extend So That its Fourier Series Will Have Only Sines 3:57 
    Find the Fourier Series for f(x) 4:19 
   Example 1 5:21 
   Example 2 8:08 
   Example 3 17:42 
   Example 4 25:13 
   Example 5 28:53 
   Example 6 42:22